Markov Skeleton Processes In The Generalized Branching Process

Branching process, as an important branch of stochastic process, is widely used. Since the classical branching processes is proposed, Branching process has developed both from simpleness to the complex, and from unitarily to multiplicity for more than one century. So, we can say the classical branching process is the foundation of other branching processes. The generating function was adopted to research the behavior of branching processes in the former research.Based on the former research results, this paper taking a particle splitting system as an example, firstly extends the Markov branching process to the generalized branching processes, and then extends the generalized branching process to the two-type branching processes; next, discusses the instantaneous distribution of the generalized branching processes and the two-type branching processes with the Markov skeleton process theory; finally, gives out the equations of instantaneous distribution of the particle splitting system. Markov skeleton process is a kind of comprehensive stochastic process, which is firstly put forward by prof.Hou zhenting and his colleagues in 1997.The process contains many classical stochastic process models, such as Markov process, semi-Markov process, piecewise-deterministic Markov process etc. They have important value in theory and application. Generalized branching process can be described as follows: in a particle splitting system, one particle does not necessarily die after splitting. It can be the same as the new born participates in the split again; Moreover, the splitting situation of all particles is independent of each other; and the splitting process is time-homogeneous; At the same time, the splitting situation of various particles is related with their history; That is to say, the individual splitting time is not subject to the negative exponential distribution but the general distribution.